Английская Википедия:Anscombe transform
In statistics, the Anscombe transform, named after Francis Anscombe, is a variance-stabilizing transformation that transforms a random variable with a Poisson distribution into one with an approximately standard Gaussian distribution. The Anscombe transform is widely used in photon-limited imaging (astronomy, X-ray) where images naturally follow the Poisson law. The Anscombe transform is usually used to pre-process the data in order to make the standard deviation approximately constant. Then denoising algorithms designed for the framework of additive white Gaussian noise are used; the final estimate is then obtained by applying an inverse Anscombe transformation to the denoised data.
Definition
For the Poisson distribution the mean <math>m</math> and variance <math>v</math> are not independent: <math>m = v</math>. The Anscombe transform[1]
- <math>A:x \mapsto 2 \sqrt{x + \tfrac{3}{8}} \, </math>
aims at transforming the data so that the variance is set approximately 1 for large enough mean; for mean zero, the variance is still zero.
It transforms Poissonian data <math>x</math> (with mean <math>m</math>) to approximately Gaussian data of mean <math>2\sqrt{m + \tfrac{3}{8}} - \tfrac{1}{4 \, m^{1/2}} + O\left(\tfrac{1}{m^{3/2}}\right)</math> and standard deviation <math> 1 + O\left(\tfrac{1}{m^2}\right)</math>. This approximation gets more accurate for larger <math>m</math>,[2] as can be also seen in the figure.
For a transformed variable of the form <math>2 \sqrt{x + c}</math>, the expression for the variance has an additional term <math>\frac{\tfrac{3}{8} -c}{m}</math>; it is reduced to zero at <math>c = \tfrac{3}{8}</math>, which is exactly the reason why this value was picked.
Inversion
When the Anscombe transform is used in denoising (i.e. when the goal is to obtain from <math>x</math> an estimate of <math>m</math>), its inverse transform is also needed in order to return the variance-stabilized and denoised data <math>y</math> to the original range. Applying the algebraic inverse
- <math>A^{-1}:y \mapsto \left( \frac{y}{2} \right)^2 - \frac{3}{8} </math>
usually introduces undesired bias to the estimate of the mean <math>m</math>, because the forward square-root transform is not linear. Sometimes using the asymptotically unbiased inverse[1]
- <math>y \mapsto \left( \frac{y}{2} \right)^2 - \frac{1}{8} </math>
mitigates the issue of bias, but this is not the case in photon-limited imaging, for which the exact unbiased inverse given by the implicit mapping[3]
- <math> \operatorname{E} \left[ 2\sqrt{x+\tfrac{3}{8}} \mid m \right] = 2 \sum_{x=0}^{+\infty} \left( \sqrt{x+\tfrac{3}{8}} \cdot \frac{m^x e^{-m}}{x!} \right) \mapsto m </math>
should be used. A closed-form approximation of this exact unbiased inverse is[4]
- <math>y \mapsto \frac{1}{4} y^2 - \frac{1}{8} + \frac{1}{4} \sqrt{\frac{3}{2}} y^{-1} - \frac{11}{8} y^{-2} + \frac{5}{8} \sqrt{\frac{3}{2}} y^{-3}.</math>
Alternatives
There are many other possible variance-stabilizing transformations for the Poisson distribution. Bar-Lev and Enis report[2] a family of such transformations which includes the Anscombe transform. Another member of the family is the Freeman-Tukey transformation[5]
- <math>A:x \mapsto \sqrt{x+1}+\sqrt{x}. \, </math>
A simplified transformation, obtained as the primitive of the reciprocal of the standard deviation of the data, is
- <math>A:x \mapsto 2\sqrt{x} \, </math>
which, while it is not quite so good at stabilizing the variance, has the advantage of being more easily understood. Indeed, from the delta method,
<math> V[2\sqrt{x}] \approx \left(\frac{d (2\sqrt{m})}{d m} \right)^2 V[x] = \left(\frac{1}{\sqrt{m}} \right)^2 m = 1 </math>.
Generalization
While the Anscombe transform is appropriate for pure Poisson data, in many applications the data presents also an additive Gaussian component. These cases are treated by a Generalized Anscombe transform[6] and its asymptotically unbiased or exact unbiased inverses.[7]
See also
References
Further reading
